EMFCSC INTERNATIONAL SCHOOL OF SUBNUCLEAR PHYSICS54th Course: The New Physics Frontiers in the LHC-2 Era

Composite particles viewed as deformed

oscillators, deformed Bose gas models and

some applications

Yu. A. Mishchenko1, A. M. Gavrilik1, I. I. Kachurik2

1Bogolyubov Institute for Theoretical Physics of NAS of Ukraine, Kyiv, Ukraine2Khmelnytskyi National University, Khmelnytskyi, Ukraine

Erice-Sicily, 14 � 23 June 2016

Plan

1. Motivation for realization (modelling) of composite particles bydeformed oscillators.

2. Operator realization of two-particle composite bosons by de-formed oscillators (def. bosons).

3. Realization for �def. boson+ fermion� composite fermi-particles (CFs). General solution for non-deformed constituents,and some particular ones.

4. Composite boson (quasiboson) as entangled bipartite system.The measures (characteristics) of entanglement between con-stituents of composite boson. Link with deformation parameterand energy dependence.

5. Application of deformed Bose gas models to e�ective accountfor the interaction/compositeness of particles: deformed virialexpansion of EOS.

6. Momentum correlation function intercept of 2nd order for ��, 𝑞-Bose gas model.

7. Conclusions.

Composite (quasi)particles in diverse branches of physics∙ Mesons, baryons, nuclei in nuclear or subnuclear physics;∙ Excitons, biexcitons, dropletons in semiconductors and nanostruc-tures (quantum dots etc.);∙ Trions (�2 electrons + hole� or �2 holes + electron�) in semicon-ductors;∙ Cooper pairs in superconductors, in the study of electron transport;∙ Bipolarons in crystals, organic semiconductors;∙ Composite fermions (�electron + magnetic �ux quanta� boundstates [Jain]) appearing in fractional quantum Hall e�ect;∙ Biphotons in quantum optics, quantum information;∙ Biphonons, triphonons, multiphonons in crystals;∙ Atoms, molecules, ... .

Deformed oscillators � nonlinear generalization of ordinaryquantum oscillator de�ned by deformation structure function 𝜙 and

𝑎†𝑎 = 𝜙(𝑁), 𝑎𝑎† = 𝜙(𝑁 + 1), [𝑁, 𝑎†] = 𝑎†, [𝑁, 𝑎] = −𝑎,

with [𝑎, 𝑎†] = 1 + 𝛿𝜙(𝑁) where 𝛿𝜙(𝑁) ≡ 𝜙(𝑁+1)−𝜙(𝑁)−1.Fock space: |𝑛⟩ = 1√

𝜙(𝑛)!(𝑎†)𝑛|0⟩, 𝜙(𝑛)! ≡ 𝜙(𝑛)·...·𝜙(1),

𝑎†|𝑛⟩ =√

𝜙(𝑛+1)|𝑛+1⟩, 𝑎|𝑛⟩ =√𝜙(𝑛)|𝑛−1⟩, 𝑁 |𝑛⟩ = 𝑛|𝑛⟩.

Motivational considerationsComposite particles

Creation& annihilation operators:

𝐴†𝛼 =

∑𝜇𝜈

Φ𝜇𝜈𝛼 𝑎†𝜇𝑏

†𝜈 ,

𝐴𝛼 =∑

𝜇𝜈Φ𝜇𝜈𝛼 𝑏𝜈𝑎𝜇,

𝑁𝛼 = 𝑁𝛼(𝑎†𝜇, 𝑎𝜇, 𝑏

†𝜈 , 𝑏𝜈).

Commutator:

[𝐴𝛼, 𝐴†𝛽] = 𝛿𝛼𝛽 −Δ𝛼𝛽(Φ

𝜇𝜈𝛾 ).

Deformed particlesGeneralized oscillator algebra:⎧⎪⎨⎪⎩𝒜†

𝛼𝒜𝛼 = 𝜑(𝒩𝛼),

[𝒜𝛼,𝒜†𝛼] = 𝜑(𝒩𝛼 + 1)− 𝜑(𝒩𝛼),

[𝒩𝛼,𝒜†𝛼] = 𝒜†

𝛼, [𝒩𝛼,𝒜𝛼] = −𝒜𝛼.

Example.Arik-Coon deformation:[𝒜𝛼,𝒜†

𝛼] = 1− (𝑞 − 1)𝒜†𝛼𝒜𝛼,

𝜑(𝒩𝛼) =𝑞𝒩𝛼 − 1

𝑞 − 1.

The realization/modelling of composite bosons (e.g. mesons, ex-citons, cooperons etc.) by deformed oscillators allows to:

I considerably simplify the calculations;I abstract away from the internal structure details;I apply well developed theory of deformed oscillators.

So, the operators for the system of composite particles map onto thedeformed oscillator operators. The internal structure information forsuch particles enters deformation parameter(s).

I. Operator realization of quasibosons by deformed bosonsComposite (or quasi-) boson creation/annihilation operators 𝐴†

𝛼, 𝐴𝛼

(mode 𝛼) are realized on the states by certain def. oscillator operators

𝐴†𝛼 =

∑𝜇𝜈

Φ𝜇𝜈𝛼 𝑎†𝜇𝑏

†𝜈 → 𝒜†

𝛼 , 𝐴𝛼 =∑

𝜇𝜈Φ𝜇𝜈𝛼 𝑏𝜈𝑎𝜇 → 𝒜𝛼 , (1)

where 𝒜†𝛼, 𝒜𝛼 are def. oscillator creation/annihilation operators:

𝒜†𝛼𝒜𝛼=𝜙(𝒩𝛼), [𝒜𝛼,𝒜†

𝛽]=𝛿𝛼𝛽(𝜙(𝒩𝛼+1)−𝜙(𝒩𝛼)

)≡𝛿𝛼𝛽(1+Δ𝜙(𝒩𝛼)),

Φ𝜇𝜈𝛼 is the composite's wavefunction, and constituent operators

𝑎𝜇, 𝑏𝜈 satisfy fermionic (bosonic) commut. relations. � We look

for such operators 𝐴𝛼, 𝐴†𝛼, 𝑁𝛼 ≡ 𝜑−1(𝐴†

𝛼𝐴𝛼), which behave on a

subspace of all quasiboson states as the ones for a system of inde-pendent deformed oscillators with structure function 𝜑(𝑁):

[𝐴𝛼, 𝐴†𝛽] ≡ −𝜖Δ𝛼𝛽 = 0 for 𝛼 = 𝛽,

[𝑁𝛼, 𝐴†𝛼] = 𝐴†

𝛼, [𝑁𝛼, 𝐴𝛼] = −𝐴𝛼,

[𝐴𝛼, 𝐴†𝛽] = 𝛿𝛼𝛽 − 𝜖Δ𝛼𝛽 = 𝜑(𝑁𝛼 + 1)− 𝜑(𝑁𝛼),

(2)

with Δ𝛼𝛽 ≡∑

𝜇𝜇′(Φ𝛽Φ

†𝛼)

𝜇′𝜇𝑎†𝜇′𝑎𝜇 +∑

𝜈𝜈′(Φ†

𝛼Φ𝛽)𝜈𝜈′𝑏†𝜈′𝑏𝜈 ↔ Δ𝜙(𝒩𝛼).,

𝜖 = +1/− 1 − for boson/fermion constituents respectively.

Remark that closed-form relation holds

[[𝐴𝛼,𝐴†𝛽],𝐴

†𝛾 ]=−𝜖

∑𝛿𝐶𝛿𝛼𝛽𝛾(Φ)𝐴

†𝛿,

where 𝐶𝛿𝛼𝛽𝛾 depend on wavefunctions Φ.

Realization conditions (2) reduce to equations for Φ𝛼 and 𝜙(𝑛):

Φ𝛽Φ†𝛼Φ𝛾 +Φ𝛾Φ

†𝛼Φ𝛽 = 0, 𝛼 = 𝛽, (3)

Φ𝛼Φ†𝛼Φ𝛼 =

𝑓

2Φ𝛼,

𝑓

2= 1− 1

2𝜑(2), (4)

𝜑(𝑛+ 1) =∑𝑛

𝑘=0(−1)𝑛−𝑘𝐶𝑘

𝑛+1𝜑(𝑘), 𝑛 ≥ 2. (5)

Their solution yields:∙ deformation structure function 𝜙𝑓 (𝑁) = (1 + 𝜖𝑓2 )𝑁 − 𝜖𝑓2𝑁

2

with∙ deformation parameter 𝑓= 2

𝑚 , 𝑚 ∈ N (𝑚 is positive integer);∙ matrices

Φ𝛼=𝑈1(𝑑𝑎)diag{0..0,

√𝑓/2𝑈𝛼(𝑚), 0..0

}𝑈 †2(𝑑𝑏), (6)

where 𝑈1(𝑑𝑎), 𝑈2(𝑑𝑏), 𝑈𝛼(𝑚) are arbitrary unitary matrices of di-mensions 𝑑𝑎 × 𝑑𝑎, 𝑑𝑏 × 𝑑𝑏 and 𝑚×𝑚. 𝜖 = +1 or −1 for fermionicresp. bosonic constituents.

Generalization to quasibosons, composed of 𝑞-fermionsCommutation relations for the constituent 𝑞-fermions:

𝑎𝜇𝑎†𝜇′ + 𝑞𝛿𝜇𝜇′𝑎†𝜇′𝑎𝜇 = 𝛿𝜇𝜇′ , 𝑏𝜈𝑏

†𝜈′ + 𝑞𝛿𝜈𝜈′ 𝑏†𝜈′𝑏𝜈 = 𝛿𝜈𝜈′ , (7)

𝑎𝜇𝑎𝜇′ + 𝑎𝜇′𝑎𝜇 = 0, 𝜇 = 𝜇′, 𝑏𝜈𝑏𝜈′ + 𝑏𝜈′𝑏𝜈 = 0, 𝜈 = 𝜈 ′. (8)

The nilpotency is absent (as opposed to the previous case):

𝑞 < 1 ⇒ (𝑎†𝜇)𝑘 = 0, (𝑏†𝜈)

𝑘 = 0, 𝑘 ≥ 2. (9)

Solving the conditions analogous to (2) we obtain:

I Resulting expression for the structure function:

𝜑(𝑛) = ([𝑛]−𝑞)2 =

(1− (−𝑞)𝑛

1 + 𝑞

)2, 𝑞 < 1. (10)

I Solution for matrices Φ𝛼 at 𝑞 < 1:

Φ𝜇𝜈𝛼 = Φ𝜇0(𝛼)𝜈0(𝛼)

𝛼 𝛿𝜇𝜇0(𝛼)𝛿𝜈𝜈0(𝛼), |Φ𝜇0(𝛼)𝜈0(𝛼)𝛼 | = 1. (11)

For more details see:[1] A. M. Gavrilik, I. I. Kachurik, Yu. A. Mishchenko, Ukr. J. Phys. 56,

948 (2011).

[2] A. M. Gavrilik, I. I. Kachurik, Yu. A. Mishchenko, J. Phys. A: Math.

Theor. 44, 475303 (2011).

Realization of “def. boson+fermion” composite fermionsCFs' creation/annihilation operators are given by �ansatz� (1) with

𝑎†𝜇, 𝑎𝜇 resp. 𝑏†𝜈 , 𝑏𝜈 � creation/annihilation operators for constituentbosons (deformed or not) resp. fermions such that

𝑎†𝜇𝑎𝜇 = 𝜒(𝑛𝑎𝜇), [𝑎𝜇, 𝑎

†𝜇′ ] = 𝛿𝜇𝜇′

(𝜒(𝑛𝑎

𝜇+1)−𝜒(𝑛𝑎𝜇)); [𝑎†𝜇, 𝑎

†𝜇′ ] = 0.

Fermionic nilpotency holds (𝐴†𝛼)2 = 0. For nondeformed constituent

boson (𝜒(𝑛) ≡ 𝑛) the anticommutator yields

{𝐴𝛼, 𝐴†𝛽} = 𝛿𝛼𝛽 +

∑𝜇𝜇′

(Φ𝛽Φ†𝛼)

𝜇′𝜇𝑎†𝜇′𝑎𝜇−∑

𝜈𝜈′(Φ†

𝛼Φ𝛽)𝜈𝜈′𝑏†𝜈′𝑏𝜈 ,

So, the concerned CFs' operators are realized by fermion operators.The validity of the realization on one-CF states yields

(Φ𝛽Φ†𝛼Φ𝛾)

𝜇𝜈 − (Φ𝛾Φ†𝛼Φ𝛽)

𝜇𝜈+

+(𝜒(2)− 2

)[(Φ𝛽Φ

†𝛼)

𝜇𝜇Φ𝜇𝜈𝛾 − (Φ𝛾Φ

†𝛼)

𝜇𝜇Φ𝜇𝜈𝛽

]= 0. (12)

For non-deformed constituent boson realization conditions (12), ...reduce to {

Tr(Φ𝛽Φ†𝛼) = 𝛿𝛼𝛽,

Φ𝛽Φ†𝛼Φ𝛾 − Φ𝛾Φ

†𝛼Φ𝛽 = 0.

(13)

a) Single CF mode 𝛼 case. General solution:

Φ𝛼 = 𝑈𝛼 diag{𝜆(𝛼)1 , 𝜆

(𝛼)2 , ...}𝑉 †

𝛼

with 𝜆(𝛼)𝑖 ≥ 0,

∑𝑖(𝜆

(𝛼)𝑖 )2 = 1, and arbitrary unitary 𝑈𝛼, 𝑉𝛼.

General remark. The number of modes for realized compositefermions 𝐷𝐶𝐹 and the constituent fermions 𝐷𝑓 satisfy: 𝐷𝐶𝐹 ≤ 𝐷𝑓 .b) CFs in 2 modes & non-deformed constituents in 3 modes.

The parametrization of two orthonormal vectors (𝜆(𝛼)1 , 𝜆

(𝛼)2 , 𝜆

(𝛼)3 ),

𝛼 = 1, 2 follows from 𝑆𝑈(3) parametrization, e.g.

𝜆(1)1 = cos 𝜃1 cos 𝜃2, 𝜆

(2)2 = cos 𝜃1 sin 𝜃2, 𝜆

(3)3 = sin 𝜃1;

and 𝜆(2)1 = − sin 𝜃1 cos 𝜃2 cos 𝜃3 − sin 𝜃2 sin 𝜃3𝑒

𝑖𝛾 ,

𝜆(2)2 = cos 𝜃2 sin 𝜃3𝑒

𝑖𝛾 − sin 𝜃1 sin 𝜃2 cos 𝜃3, (14)

𝜆(2)3 = cos 𝜃1 cos 𝜃3, 0 ≤ 𝜃1, 𝜃2, 𝜃3 ≤ 𝜋/2, 0 ≤ 𝛾 ≤ 2𝜋.

Solution for any number of modes (non-def. CF constituents).

Φ𝛼 = 𝑈 diag{𝜆(𝛼)1 , 𝜆

(𝛼)2 , ...}𝑉 †

where 𝑈 , 𝑉 are �xed (for any 𝛼) unitary matrices, 𝜆(𝛼) =

(𝜆(𝛼)1 , 𝜆

(𝛼)2 , ...) are complex orthonormal vectors. Compare with (6).

Interpretation of the involved parameters The concerned pa-rameters 𝜃𝑖, 𝑖 = 1, 3, and 𝛾 (the 3-mode case) should correspondto CF internal quantum numbers like spin, the ones determining CFbinding energy, etc.

Possible applications: trions, baryons. The results of thiswork involving nontrivial deformation 𝜒 may be applied to the ef-fective description of tripartite (three-component) composite parti-cles e.g. when two constituents form a bound state modeled by𝜒-deformed boson. For such modeling of composite constituent bo-son, the quasibosons realization could be relevant. Proper combina-tion of the two realizations, quasibosonic and CF ones, can providean alternative e�ective description of tripartite complexes like trions(e.g. exciton-electron or �electron + electron + hole� composites)or baryons (viewed either as three quark bound state, or as quark-diquark composite particles) signi�cantly simpler than their directtreatment. These issues deserve special detailed study.

II. Entanglement in composite boson vs deformationparameter [A. Gavrilik, Yu. Mishchenko, PLA 376, 1596 (2012)]

The state of the quasiboson which can be realized by deformedoscillator is entangled (inter-component entanglement):

|Ψ𝛼⟩=∑𝑚

𝑘=1

1√𝑚|𝑣𝛼𝑘 ⟩⊗|𝑤𝛼

𝑘 ⟩, |𝑣𝛼𝑘 ⟩=𝑈𝜇𝑘1 |𝑎𝜇⟩, |𝑤𝛼

𝑘 ⟩= ��𝑘𝜈2 |𝑏𝜈⟩,

𝜆𝛼𝑘 = 𝜆 =

√𝑓/2 =

√1/𝑚. (15)

Calculation of the entanglement characteristics yields:I Schmidt rank 𝑟 = 𝑚;I Schmidt number (𝑃 � the purity of subsystems)

𝐾 =

[∑𝑘(𝜆𝛼

𝑘 )4

]−1

= 1/𝑃 = 𝑚; (16)

I Entanglement entropy 𝑆entang = −∑

𝑘(𝜆𝛼

𝑘 )2 ln(𝜆𝛼

𝑘 )2= ln(𝑚);

I Concurrence 𝐶 =

[𝑟

𝑟 − 1

(1−

∑𝑘(𝜆𝛼

𝑘 )4)]1/2

= 1.

Remark. Strongly entangled composite boson (high 𝑚) ap-proaches standard boson at small quantum numbers 𝑛:

𝜑(𝑛) ≈ 𝜑𝑏𝑜𝑠𝑜𝑛(𝑛) ≡ 𝑛, 𝑛 ≪ 𝑚, 𝑚 ≫ 1.

Generalization to multi-quasiboson statesExample 1. Multi-quasiboson state, one mode

|𝑛𝛼⟩ = [𝜑(𝑛𝛼)!]−1/2(𝐴†

𝛼)𝑛𝛼 |0⟩ (17)

(𝜑-factorial 𝜑(𝑛)!𝑑𝑒𝑓= 𝜑(1) · ... · 𝜑(𝑛)).

Entanglement characteristics for (17):

𝐾𝜖=+1=𝐶𝑚𝑛𝛼 , 𝐾𝜖=−1=𝐶𝑛𝛼

𝑚+𝑛𝛼−1;

𝑆entang|𝜖=+1=ln𝐶𝑚𝑛𝛼 , 𝑆entang|𝜖=−1=ln𝐶𝑛𝛼

𝑚+𝑛𝛼−1.

Example 2. 𝑛-quasiboson Fock states with 1 quasiboson per mode:

|Ψ⟩ = 𝐴†𝛾1 · ... ·𝐴

†𝛾𝑛 |0⟩, 𝛾𝑖 = 𝛾𝑗 , 𝑖 = 𝑗, 𝑖, 𝑗 = 1, ..., 𝑛.

Entanglement characteristics:𝐾𝜖=+1 = 𝐾𝜖=−1 = 𝑚𝑛;

𝑆entang|𝜖=+1 = 𝑆entang|𝜖=−1 = 𝑛 ln(𝑚).

Example 3. For the coherent state (two bosonic constituents)

|Ψ𝛼⟩=𝐶(𝒜;𝑚)∑∞

𝑛=0

𝒜𝑛

𝜑(𝑛)!(𝐴†

𝛼)𝑛|0⟩, 𝐴𝛼|Ψ𝛼⟩ = 𝒜𝛼|Ψ𝛼⟩

Schmidt number and Entanglement entropy were also calculated.

CFs: three modes 𝜇, 𝜈 = 1, 3 for constituents:

Figure 1: Left: Equi-entropic curves (for constant CF entanglement entropy

𝑆(𝛼)ent ) versus parameters 𝜃1, 𝜃2, at a fixed mode 𝛼 = 1 or 2.

Right: Entanglement entropy 𝑆(2)ent(𝜃

(2)1 , 𝛾′), 𝛾′ ≡ arg(𝜆

(1)1 𝜆

(1)2 𝜑11𝜑22), for

a CF in 𝛼 = 2 mode at fixed entanglement entropy 𝑆(1)ent = ln 3 for 𝛼 = 1

mode of CF.

Energy dependence of entanglement entropy for thestates of composite boson (quasiboson) systems

[A.M. Gavrilik, Yu.A. Mishchenko, J. Phys. A 46, 145301 (2013)]

Entanglement characteristics between the constituents of a quasi-boson and their energy dependence are important in quantum infor-mation research: in quantum communication, entanglement produc-tion/enhancement, quantum dissociation processes, particle additionor subtraction in general and in the teleportation problem, etc.We take the Hamiltonian of deformed oscillators system as

𝐻 =1

2

∑𝛼

~𝜔𝛼

(𝜙(𝑁𝛼) + 𝜙(𝑁𝛼 + 1)

). (18)

Single quasiboson case

𝑆ent(𝐸) = ln𝜖

32 − 𝐸

~𝜔=

⎧⎪⎨⎪⎩− ln

(32− 𝐸

~𝜔

), 𝜖 = 1,

1

2≤ 𝐸

~𝜔≤ 3

2,

− ln( 𝐸

~𝜔− 3

2

), 𝜖 = −1,

3

2≤ 𝐸

~𝜔≤ 5

2.

The corresponding plots are presented on Fig. 2 and Fig. 3.

Figure 2: Dependence of the entan-glement entropy 𝑆ent on the energy𝐸𝛼 for a single composite boson inthe case of fermionic componentsi.e. at 𝜖 = +1.

Figure 3: Dependence of the entan-glement entropy 𝑆ent on the energy𝐸𝛼 for a single composite boson inthe case of bosonic components i.e.at 𝜖 = −1.

Hydrogen atom viewed as quasiboson. � Cannot be realizedby quadratically deformed oscillators. The creation operator for hy-drogen atom1 with zero total momentum and quantum number 𝑛

𝐴†0𝑛 =

(2𝜋~)3/2√𝑉

∑p

𝜑p𝑛𝑎(𝑒)†p 𝑏

(𝑝)†−p , (19)

where 𝑎(𝑒)†p and 𝑏

(𝑝)†−p are the creation operators for electron and

proton respectively taken with opposite momenta. The momentum-space wavefunction 𝜑p𝑛 is determined by the Schrodinger equation:

𝜑p𝑛=

∫𝑒

𝑖~pr

(2𝜋~)3/2𝜑𝑛(r)𝑑

3r; −~2∇2

2𝑚𝜑𝑛(r)+𝑈(r)𝜑𝑛(r)=𝐸𝑛𝜑𝑛(r).

Expansion (19) can be viewed directly as the Schmidt decomposition

for the state 𝐴†0𝑛|0⟩ with Schmidt coe�cients 𝜆p = (2𝜋~)3/2√

𝑉𝜑p𝑛.

Then the entanglement entropy for the hydrogen atom is given by

𝑆ent = −∑p

|𝜆p|2 ln |𝜆p|2 = −∑p

(2𝜋~)3

𝑉|𝜑p𝑛|2 ln

((2𝜋~)3𝑉

|𝜑p𝑛|2).

1Note that similar ansatz is used for the excitonic creation operators

Let us consider the simplest case of quantum numbers 𝑙=0 & 𝑚=0.

Figure 4: Dependence of the entanglement entropy Δ𝑆 = 𝑆ent − 𝑆(0)ent on the

energy 𝐸 for Hydrogen atom.

III. Deformed Bose gas models. Preliminaries

Approach I. Standard relations apply to deformed physical quantities(dependent on deformation parameter(s) 𝑞):

𝐹𝑖 → 𝐹(𝑞)𝑖 , 𝑅(𝐹𝑖) = 0 → 𝑅(𝐹

(𝑞)𝑖 ) = 0. (20)

In [N. Swamy, 2009] 𝑞-Bose gas model involves the deformed parti-

cle number 𝑁 (𝑞)=𝑧𝒟(𝑞)𝑧 ln𝑍(0), obtained using deformed derivative

𝒟(𝑞)𝑧 . Then, other thermodyn. quantities can be derived. E.g., 𝑞-

deformed virial expansion 𝑃𝑣𝑘B𝑇

=∑∞

𝑘=1𝑎𝑘(𝜖)(𝜆3/𝑣)𝑘−1, 𝜖 ≡ 𝑞 − 1,

along with a few virial coe�cients 𝑎𝑘(𝜖) was found.Approach II. Deform defining relations for a physical system (saycommutation rel-ns): 𝑅(𝐹𝑖) = 0 → 𝑅𝑞(𝐹𝑖) = 0.For instance, deformed (nonlinear) oscillator is de�ned by deforma-tion structure function 𝜙 and the relations:

𝑎†𝑎 = 𝜙(𝑁), 𝑎𝑎† = 𝜙(𝑁+1), [𝑁, 𝑎†] = 𝑎†, [𝑁, 𝑎] = −𝑎,

then [𝑎, 𝑎†] = 1 + 𝛿𝜙(𝑁), 𝛿𝜙(𝑁) ≡ 𝜙(𝑁+1)−𝜙(𝑁)−1.

Deformation of thermodynamics rel-s using 𝜙-derivativeIn 𝜙-Bose gas model def. number of particles (𝑧=𝑒𝛽𝜇 � fugacity)

𝑁 (𝜙)=𝜙(𝑧𝑑

𝑑𝑧

)ln𝑍(0) ≡𝑧𝒟(𝜙)

𝑧 ln𝑍(0)=𝑉

𝜆3

∑∞

𝑛=1𝜙(𝑛)

𝑧𝑛

𝑛5/2,

where ln𝑍(0) = −∑

𝑖 ln(1 − 𝑧𝑒−𝛽𝜀𝑖) and the 𝜙-derivative 𝒟(𝜙)𝑧 is

used (analog of Jackson's 𝑞-derivative): 𝑧 𝑑𝑑𝑧 → 𝑧𝒟(𝜙)

𝑧 = 𝜙(𝑧 𝑑𝑑𝑧

).

Recovering def. partition function 𝑍(𝜙) from 𝑁 (𝜙) =(𝑧 𝑑𝑑𝑧

)ln𝑍(𝜙)

we obtain 𝜙-deformed virial (𝜆3/𝑣)-expansion (let 𝜙(1)=1)

𝑃 (𝜙)

𝑘B𝑇=

(𝑣(𝜙)

)−1{∑∞

𝑘=1𝑉

(𝜙)𝑘

( 𝜆3

𝑣(𝜙)

)𝑘−1}=

(𝑣(𝜙)

)−1{1−𝜙(2)

27/2𝜆3

𝑣(𝜙)+

+(𝜙(2)2

25−2𝜙(3)

37/2

)( 𝜆3

𝑣(𝜙)

)2+(−3𝜙(4)

47/2+𝜙(2)𝜙(3)

25/233/2−5𝜙(2)3

217/2

)( 𝜆3

𝑣(𝜙)

)3+

+(−4𝜙(5)

57/2+𝜙(2)𝜙(4)

211/2−2𝜙(3)3

35−𝜙(2)2𝜙(3)

2333/2+7𝜙(2)4

210

)( 𝜆3

𝑣(𝜙)

)4+...

},

where 𝑣(𝜙) = 𝑉𝑁(𝜙) is speci�c volume, 𝜆 � thermal wavelength,

𝑉(𝜙)𝑘 , 𝑘 = 1, 2, ..., are 𝜙-deformed virial coe�cients.

Second virial coefficient for a gas with interactionAs known,

𝑉2−𝑉(0)2 =−81/2

∑𝐵

𝑒−𝛽𝜀𝐵 − 81/2

𝜋

∑′

𝑙

(2𝑙+1)

∫ ∞

0𝑒−𝛽 ~2𝑘2

𝑚𝜕𝛿𝑙(𝑘)

𝜕𝑘𝑑𝑘

where 𝐵 runs over bound states, 𝑙 is the angular momentum, and𝛿𝑙(𝑘) partial wave phaseshift. In low-energy approximation we retainonly the 𝑙 = 0 summand (𝑠-wave approximation). Resp. phaseshift𝛿0(𝑘) is determined by Schrodinger eq. for a speci�ed interaction.Low-energy limit and 𝑠-wave approximation (𝑙 = 1 e�ects arenegligible). The following expansion � effective range (or “shape-

independent”) approximation holds:

𝑘 ctg 𝛿0=−1

𝑎+1

2𝑟0𝑘

2+..., 𝑟0=2

∫ ∞

0𝑑𝑟[(

1− 𝑟

𝑎

)2−𝜒2

0(𝑟)], (21)

where 𝑎 is the scattering length, 𝑟0 � e�ective range (radius), and𝜒0(𝑟) � the radial wavefunction of the lowest state multiplied by 𝑟.

⇒ 𝜕𝛿0/𝜕𝑘 = −𝑎+ (𝑎− 3𝑟0/2)𝑎2𝑘2 +𝑂(𝑘4).

For 𝑠-wave approximation we obtain

𝑉2−𝑉(0)2 =−81/2

∑𝐵

𝑒−𝛽𝜀𝐵+2𝑎

𝜆𝑇−2𝜋2

(1−3

2

𝑟0𝑎

)( 𝑎

𝜆𝑇

)3+... . (22)

∙ Hard spheres interaction potential [Pathria]:

𝑈(𝑟)=

{+∞, 𝑟<𝐷;

0, 𝑟>𝐷.⇒ 𝑉2−𝑉

(0)2 =2

𝐷

𝜆𝑇+10𝜋2

3

( 𝐷

𝜆𝑇

)5+... (𝑙=0, 2).

We also considered: constant repulsive, square-well, anomalous scat-tering, scattering resonances, modi�ed Poschl-Teller and inversepower repulsive potentials.

With listed 𝑎 and 𝑟0 the second virial coe�cient can be evalu-ated (22) for mentioned potentials of interparticle interaction.The gas of non-interacting but composite bosons. Using theanzats 𝐴†

𝛼 =∑

𝜇𝜈 Φ𝜇𝜈𝛼 𝑎†𝜇𝑏

†𝜈 and the known formula

𝑉2 =1

2!𝑉

[(Tr1 𝑒

−𝛽𝐻1)2 − Tr2 𝑒−𝛽𝐻2

]. (23)

if for all (k, 𝑛)-modes (𝐴†k,𝑛)

2|0⟩ = 0 we obtain that in the absenceof explicit interaction between composite bosons

𝑉2(𝑇 )− 𝑉(0)2 = − 1

25/2

(∑𝑛𝑒−2𝛽𝜀𝑖𝑛𝑡

𝑛 − 1). (24)

Effective account for the interaction/compositeness ofparticles by 𝑞- resp. ��-deformations

Interparticle interaction. In [N. Swamy, J.Stat.Mech (2009)] the 𝑞-

deformation given by structure function [𝑁 ]𝑞 ≡ 1−𝑞𝑁

1−𝑞 was interpretedas incorporating the e�ects of interparticle interaction. That lead to𝑞-deformed virial expansion

𝑃𝑣

𝑘B𝑇=

∑∞

𝑘=1𝑎𝑘(𝜖)

(𝜆3

𝑣

)𝑘−1, 𝜖 = 𝑞 − 1, (25)

with virial coe�cients 𝑎𝑘(𝜖). They utilized Jackson derivative 𝒟(𝑞)𝑧 .

Compositeness of particles. In [Gavrilik et al, J.Phys.A (2011)]composite (two-fermion or two-boson) quasi-bosons with cre-ation/annihilation operators (1) were realized by def. bosons withquadratic DSF 𝜙��(𝑁).Unification of ��- and 𝑞-deformations. The e�ective descriptionof the both mentioned factors may be expected from a combinationof DSFs [𝑁 ]𝑞 and 𝜙��(𝑁). The simplest (but non-unique) variant is

𝜙��,𝑞(𝑁)=𝜙��([𝑁 ]𝑞)=(1+��)[𝑁 ]𝑞−��[𝑁 ]2𝑞 (26)The respective ��, 𝑞-deformed Bose gas model was recently proposedin [Gavrilik, Mishchenko, Ukr. J. Phys., 2013].

Deformed second virial coefficient vs. microscopicallycalculated one. Consequences for def. parameters

We use the obtained deformed virial expansion based on 𝜙��,𝑞(𝑁),see (26), to juxtapose with the resp. results in microscopic descrip-tion. So, the �rst virial coe�cients are

𝑉(��,𝑞)2 = −𝜙��,𝑞(2)

27/2, 𝑉

(��,𝑞)3 =

𝜙��,𝑞(2)2

25− 2𝜙��,𝑞(3)

37/2. (27)

In our approach, parameter 𝑞 from 𝜙��,𝑞(𝑁) corresponds to e�ectiveaccount for interparticle interaction, and �� � for compositeness.Effective account for interaction up to (𝜆3/𝑣)2. Microscopictreatment yields, see (22),

𝑉2−𝑉(0)2 =−81/2

∑𝐵

𝑒−𝛽𝜀𝐵+2𝑎

𝜆𝑇−2𝜋2

(1−3

2

𝑟0𝑎

)( 𝑎

𝜆𝑇

)3+... . (28)

According to our interpretation 𝑉(��,𝑞)2 −𝑉

(0)2 |��=0=

2−𝜙��,𝑞(2)

27/2|��=0=

1−𝑞27/2

. Equating to (28) yields

𝑞=𝑞(𝑎, 𝑟0, 𝑇 )=1−29/2𝑎

𝜆𝑇+29/2𝜋2

(1−3

2

𝑟0𝑎

)( 𝑎

𝜆𝑇

)3+...+25

∑𝐵

𝑒−𝛽𝜀𝐵 ,

where 𝜆𝑇 = ℎ/√2𝜋𝑚𝑘𝐵𝑇 is thermal wavelength.

Effective account for the compositeness up to (𝜆3/𝑣)2-terms.In the absence of explicit interaction between quasibosons, cf. (24),

𝑉2(𝑇 ) = − 1

25/2

∑𝑛𝑒−2𝛽𝜀𝑖𝑛𝑡

𝑛

On the other hand, according to our interpretation 𝑉(��,𝑞)2 −𝑉 (0)

2 |𝑞=1=2−𝜙��,𝑞(2)

27/2|𝑞=1=

��25/2

. After equating,

⇒ �� = ��(𝜀𝑖𝑛𝑡𝑛 ,Φ𝜇𝜈𝛼 , 𝑇 ) = 1−

∑𝑛𝑒−2𝛽𝜀𝑖𝑛𝑡

𝑛 . (29)

Structure function 𝜙��,𝑞(𝑁) with 𝑞=𝑞(𝑎, 𝑟0, 𝑇 ), ��= ��(𝜀𝑖𝑛𝑡𝑛 ,Φ𝜇𝜈𝛼 , 𝑇 )

is chosen for e�ective account (in certain approximation) for thefactors of interaction and of composite structure of particles.* The temperature dependence of def. parameter 𝑞(. . . , 𝑇 ) appearsunexpected since in our interpretation def. parameter characterizesthe nonideality of deformed Bose gas.

III. Correlation function intercept for ��, 𝑞-Bose gas model[Nucl. Phys. B 891, 466 (2015)]

Figure 5: STAR(RHIC)-data

STAR (RHIC) experiment on 𝜋𝜋-correlations

shows that for intercept 𝜆(2)(k) ≡ 𝑛(2)k,k

(𝑛k)2− 1:

1) 𝜆(2)(𝑘) =const (depends on momentum);2) all exper. values are < 1 (for usual Bosegas 𝜆(2) = const = 1).

The ��, 𝑞-deformed Bose gas model is basedon a system of ��, 𝑞-deformed oscillators withstructure function 𝜙��,𝑞(𝑁), see (26) (just thecommut. relations are deformed). Intercept of2nd order as momentum k function is de�ned in the model as:

𝜆(2)𝜙 (k) =

⟨(𝑎†k)2(𝑎k)

2⟩⟨𝑎†k𝑎k⟩2

− 1 =⟨𝜙(𝑁k)𝜙(𝑁k − 1)⟩

⟨𝜙(𝑁k)⟩2− 1, (30)

where ⟨...⟩ is statistical average: ⟨𝐹 ⟩= Tr𝐹 exp(−𝛽𝐻)Tr exp(−𝛽𝐻) .

Hamiltonian is taken linear: 𝐻 =∑

k 𝜀k𝑁k.

Deformed analogs of one- and two-particle distributions in��, 𝑞-deformed Bose gas model

Here we calculate the deformed one- and two-particle distributionsfor ��, 𝑞-def. Bose gas model de�ned by DSF

𝜙��,𝑞(𝑁) = 𝜙��([𝑁 ]𝑞) = (1 + ��)[𝑁 ]𝑞 − ��([𝑁 ]𝑞)2 ≡ [𝑁 ]��,𝑞,

[𝑁 ]𝑞≡ 1−𝑞𝑁

1−𝑞 , with the interpretation of 𝑞 and �� as responsible resp.for the interaction [N. Swamy, 2009] and compositeness [Gavrilik et

al, 2011]. The de�nition of the distributions formally coincides withthe non-deformed ones,

𝑛k = ⟨𝑎†k𝑎k⟩, 𝑛(2)k,k′ = ⟨𝑎†k𝑎

†k′𝑎k′𝑎k⟩, (31)

but involves creation and annihilation operators, 𝑎†k resp. 𝑎k forthe 𝜙��,𝑞-def. bosons which obey the following set of commutationrelations given by DSF 𝜙��,𝑞:

[𝑁k, 𝑎†k′ ] = 𝛿kk′𝑎†k, [𝑁k, 𝑎k′ ] = −𝛿kk′𝑎k,

[𝑎k, 𝑎†k] = 𝜙��,𝑞(𝑁k + 1)− 𝜙��,𝑞(𝑁k), 𝑎†k𝑎k = 𝜙��,𝑞(𝑁k).

One- and two-particle deformed distributionsDetailed analysis of the applicability of chosen struct. function yieldstwo possibilities: discrete or continuous (��, 𝑞). In the discrete case:

⟨𝑎†k𝑎k⟩=1

𝑧−𝑞+(𝜙��,𝑞(2)−[2]𝑞)

(1− [𝑁max+1]𝑞2

[𝑁max+1]𝑧

)(𝑧−𝑞)(𝑧−𝑞2)

=1

𝑧−𝑞+𝛿𝜙��,𝑞(2)(1−𝑅)

(𝑧−𝑞)(𝑧−𝑞2),

where 𝛿𝜙��,𝑞(𝑛)≡𝜙��,𝑞(𝑛)−[𝑛]𝑞, 𝑅≡𝑅��,𝑞(𝑧)=[𝑁max+1]𝑞2

[𝑁max+1]𝑧, (32)

𝑧=𝑒𝑥, 𝑥=𝛽~𝜔k, 𝛽=(𝑘𝐵𝑇 )−1. For ⟨(𝑎†k)

2𝑎2k⟩ we �nd:

⟨(𝑎†k)2𝑎2k⟩=

𝜙��,𝑞(2)

(𝑧−𝑞)(𝑧−𝑞2)

{1 + (1−𝑅)

𝛿𝜙��,𝑞(3)(𝑧+𝑞2− 𝑞2[4]𝑞

𝜙��,𝑞(2)

)(𝑧 − 𝑞3)(𝑧 − 𝑞4)

−

−𝑞2[2]𝑞𝑅

([2]𝑞 [3]𝑞𝜙��,𝑞(2)

−2𝑞−1)(𝑧−𝑞2) + 𝑞2(𝑞−1)2

(𝑧 − 𝑞3)(𝑧 − 𝑞4)

}. (33)

In the continuous case, take the limit 𝑁max → ∞ to obtain

⟨𝑎†k𝑎k⟩ =𝑧 + 𝜙��,𝑞(2)− [3]𝑞(𝑧 − 𝑞)(𝑧 − 𝑞2)

,

⟨(𝑎†k)2𝑎2k⟩ =

𝜙��,𝑞(2)

(𝑧−𝑞)(𝑧−𝑞2)

{1+

(𝜙��,𝑞(3)−[3]𝑞)(𝑧−𝑞2

( [4]𝑞𝜙��,𝑞(2)

−1))

(𝑧 − 𝑞3)(𝑧 − 𝑞4)

}.

Intercept of two-particle correlation function andcomparison with experiment

Plugging (32)-(33) in (30) yields the resulting expression for the

intercept 𝜆(2)(k).The dependence 𝜆(2)(𝐾) on the momentum 𝐾 = |k| for some

values of ��, 𝑞 and temperature 𝑇 versus exper. data for 𝜋-mesonintercepts from RHIC/STAR is shown in Fig. 6, where we take ~𝜔 =√𝑚2 +𝐾2, and for 𝑚 the 𝜋-meson mass (139.5 MeV).

Figure 6: Intercept 𝜆(2)(𝐾) vs. mo-mentum 𝐾, for different values of 𝑞,�� and 𝑇 chosen to fit experimentaldata. Exper. dots [STAR] are shownby boxes.

Conclusions∙ For quasibosons built of 2 fermions, 2 bosons or 2 𝑞-fermionstheir operator realization by deformed oscillators (deformed bosons)with quadr. struct. function is found. The �fermion + def. boson�composite fermi-particles are also treated.

∙ The deformation parameter is established to be unambiguously de-termined by the entanglement characteristics for the realized compos-ite bosons. Thus, inter-component entanglement reveals the physicalmeaning of the deformation parameter.

∙ The relation of the 2nd vir. coe�cient of the ��, 𝑞-def. Bose gasmodel to the parameters of interaction and compositeness is found.Def. parameter 𝑞 is linked to the scattering length and effective radiusof interaction, and �� relates to internal energy levels of quasibosons.

∙ Deformed 2-particle distribution and correlation function interceptin resp. ��, 𝑞-deformed Bose gas model are explicitly obtained. Thecomparison of the obtained 2-particle correlation intercept with 2-pion intercepts from RHIC/LHC experiments is made. It shows aqualitative agreement.

Publications:

1. Gavrilik A.M., Kachurik I. I. and Mishchenko Yu. A. it Quasi-bosons composed of two 𝑞-fermions: realization by deformedoscillators. J. Phys. A: Math. Theor., 2011, 44, 475303.

2. Gavrilik A.M. and Mishchenko Yu. A. Entanglement in compos-

ite bosons realized by deformed oscillators. Phys. Lett. A,2012, 376, P. 1596�1600.

3. Gavrilik A.M. and Mishchenko Yu. A. Energy dependence of

the entanglement entropy of composite boson (quasiboson) sys-

tems. J. Phys. A: Math. Theor., 2013, 46, 145301.

4. Gavrilik A. M. and Mishchenko Yu. A. Virial coefficients in the

(��, 𝑞)-deformed Bose gas model related to compositeness of par-

ticles and their interaction: Temperature-dependence problem.Phys. Rev. E 90, 052147 (2014).

5. Gavrilik A. M. and Mishchenko Yu. A. Correlation function in-

tercepts for ��, 𝑞-deformed Bose gas model implying effective ac-

counting for interaction and compositeness of particles. Nucl.Phys. B 891, 466 (2015).

Thank you for your attention!